(1)	$\begin{array}{l}\displaystyle X_f = \frac{Q}{2 \, w_f \, h_f}\end{array}$

(2)	$\begin{array}{l}\displaystyle w_f = \frac{2 \pi \, p_{\rm net} \, h_f}{5 \, E'}\end{array}$

Net pressure at the wellbore

(3)	$\begin{array}{l}\displaystyle p_{\rm net} = \left( \frac{20}{\pi^2} \cdot \frac{E'^4 \, q \, Q \, \mu}{h_f^6} \right)^{1/5}\end{array}$

where

$\begin{array}{l}t\end{array}$	fracture fluid injection time
$\begin{array}{l}q\end{array}$	fracture fluid injection rate
$\begin{array}{l}Q(t) = q \cdot t\end{array}$	cumulative fracture fluid injection over time $\begin{array}{l}t\end{array}$
$\begin{array}{l}h_f\end{array}$	fracture height
$\begin{array}{l}E' = E \, / \, (1-\nu^2)\end{array}$	plane stress
$\begin{array}{l}E\end{array}$	Young modulus
$\begin{array}{l}\nu\end{array}$	Poisson's ratio
$\begin{array}{l}\mu\end{array}$	fracture fluid viscosity

Derivation

(4)	$\begin{array}{l}\displaystyle X_f = 0.524 \, \left( \frac{q^3 E'}{\mu \, h_f^4} \right)^{1/5} \, t^{4/5} = 0.524 \, \left( \frac{E'}{\mu \, h_f^4} \frac{Q^4}{q}\right)^{1/5} = 0.8 \cdot \frac{E' \, Q}{p_{\rm net} \, h_f^2} = \frac{Q}{w_f \, h_f}\end{array}$

(5)	$\begin{array}{l}\displaystyle w_{f0} = 3.04 \, \left( \frac{q^2 \mu}{E' \, h_f} \right)^{1/5} \, t^{1/5}= 3.04 \, \left( \frac{q \, Q \mu}{E' \, h_f} \right)^{1/5} = \frac{2 \, p_{\rm net} \, h_f}{E'}\end{array}$

(6)	$\begin{array}{l}\displaystyle w_f = \frac{\pi}{5} \, w_{f0} = \frac{2 \pi \, p_{\rm net} \, h_f}{5 \, E'}\end{array}$

Net pressure at the wellbore

(7)	$\begin{array}{l}\displaystyle p_{\rm net} = 1.524 \, \left( \frac{E'^4 \, q^2 \, \mu}{h_f^6} \right)^{1/5} \, t^{1/5} = 1.524 \, \left( \frac{E'^4 \, q \, Q \, \mu}{h_f^6} \right)^{1/5}\end{array}$

where

$\begin{array}{l}t = Q(t)/q\end{array}$	injection time
$\begin{array}{l}q\end{array}$	injection rate
$\begin{array}{l}Q(t)\end{array}$	cumulative injection over time $\begin{array}{l}t\end{array}$
$\begin{array}{l}h_f\end{array}$	fracture height
$\begin{array}{l}E' =\frac{E}{1-\nu^2}\end{array}$	plain stress
$\begin{array}{l}E\end{array}$	Young modulus
$\begin{array}{l}\nu\end{array}$	Poisson ratio
$\begin{array}{l}\mu\end{array}$	fluid viscosity

Reference

Perkins, Kern and Nordgren

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